Liquid-gas phase transition in the canonical ensemble of asymmetric nuclear matter

Transcription

1 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter K. Myazak E-mal: Abstract New calculus of the lqud-gas hase trasto s develoed for the caocal esemble of asymmetrc uclear matter. I cotrast to the famlar geometrcal costructo, the ressure ad the chemcal otetals are determed for de te values of the baryo destes of roto ad eutro ad the temerature. Because the dervatves of chemcal otetals by ressure have dscotutes at the eds of the hase trasto, we coclude agast the recedg works that the hase trasto s the rst order. Due to the eld-deedet meso-ucleo coulg costats our relatvstc mea- eld model of uclear matter, the secto of bodal surface has o crtcal ot. The aearace of the retrograde codesato s also roved. The lqud-gas hase trasto warm uclear matter [1,2] s a recet toc uclear hyscs. It s exermetally vestgated [3,4] the multfragmetato reactos of heavy os. The uclear matter roduced the reacto s charge asymmetrc. It s a bary system that has two deedet chemcal otetals of roto ad eutro. Based o the Gbbs codto the hase equlbrum, both the chemcal otetals gaseous ad lqud hases are equlbrated. Referece [5] determed them terms of the geometrcal costructo, whch was also used the other works [6-9] of the lqud-gas hase trasto asymmetrc uclear matter. I the geometrcal costructo, the chemcal otetals are determed for a xed value of the ressure ad the asymmetry of uclear matter. However, the object s ether a caocal or a grad-caocal esemble. I cotrast to the codesed matter hyscs we caot ress uclear matter uder a de te ressure uclear exermets. I ths ot of vew, the geometrcal costructo s ot hyscally cosstet wth the lqud-gas hase trasto uclear multfragmetato reactos. O the other had, aother strategy s roosed recetly Ref. [1], whch cosders a eutro gradcaocal but roto caocal esemble. The vestgato of hase equlbrum s much easer tha the geometrcal costructo because the Maxwell costructo s alcable. Nevertheless, the total system Ref. [1] s ether caocal or grad-caocal yet. I the reset aer we vestgate for the rst tme the lqud-gas hase trasto the caocal esemble of bary uclear system. We are based o the relatvstc mea- eld (RMF) model of asymmetrc uclear matter develoed Refs. [11] ad [12]. The 1

2 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter model has advatages over the other RMF models Refs. [5-8]. It ca reroduce the recet astroomcal observatos of NSs, the desty-deedece of the uclear symmetry eergy ad the crtcal temerature of symmetrc uclear matter. The successes are essetally due to the eld-deedet e ectve N N X (X =,!, ad ) coulg costats the model: g ()X = 1 2 h (1 + X ) + (1 X ) m() 2 v() 2 g NNX ; (1) where M = m s the e ectve mass ad V = v s the vector otetal of a roto or a eutro the medum. We assume =! = 2=3 ad = =. The coulg costats g NNX are determed Ref. [11]. At te temerature T the thermodyamc otetal er volume ~ =V s ~ = 1 2 m2 h m2 h m2! h! m2 h 3 2 k B T X Z 1 =; d 3 k (2) 3 l 1 + ex E k k B T Ek + l 1 + ex ; k B T where k B s the Boltzma costat ad E k = (k2 + M 2 ) 1=2. The s-sos degeeracy s = 2. The s de ed usg the chemcal otetal ad the vector otetal V of a roto or a eutro as (2) = V : (3) The e ectve masses ad the vector otetals are determed by extremzg ~ = S + h! h + h h m h h m = ; ~ V B + h + h h v v h h v = ; = S + h + h h m h!! h m h m = ; (6)

3 K. V = B + h! h + h h h! h h v = ; (7) where the mea- elds are calculated [11] from the e ectve masses ad the vector otetals. The exlct exressos of the dervatves of mea elds are also gve Ref. [11]. The baryo ad scalar destes are de ed by B = S = Z 1 Z 1 d 3 k (2) 3 [ k (m ; v ; ; T ) k (m ; v ; ; T )]; (8) d 3 k (2) 3 M E k [ k (m ; v ; ; T ) + k (m ; v ; ; T )]: (9) The Ferm-Drac dstrbuto fuctos of ucleo ad atucleo are E k (m 1 ; v ; ; T ) = 1 + ex k ; (1) k B T k (m ; v ; ; T ) = E ex k + : (11) k B T I the followg umercal aalyses we have erformed the Ferm tegral drectly usg the adatve automatc tegrato wth 2-ots Gaussa quadrature. As metoed above we cosder the caocal esemble of asymmetrc uclear matter. I other words we secfy the temerature T ad the baryo destes of roto ad eutro uclear matter. I lace of each baryo desty the total baryo desty ad the sos asymmetry B = B + B ; (12) B a = B ; (13) B are sec ed. The, the 6th-rak olear smultaeous equatos (4)-(7), (12) ad (13) are solved so that we have the e ectve masses, the vector otetals ad the chemcal otetals of roto ad eutro. Usg the results, the ressure s calculated by P = X Z 1 =; dk k4 [ Ek k (m ; v ; ; T ) + k (m ; v ; ; T )] 1 2 m2 h m2 h m2! h! m2 h 3 2 : (14) 3

4 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter The black curve Fg. 1 shows a sotherm the ressure-desty lae for a = :3 ad T = 13MeV. (We set the Boltzma costat as ut.) It exhbts tycal ature of va der Waals equato-of-state. The black curve Fg. 2 shows the Helmholtz eergy er baryo F=A = G=A P= B as a fucto of volume, where G=A = B + B =B s the Gbbs eergy er baryo. A cocave o the curve s a sgal of the lqud-gas hase trasto. Because the hyscal Helmholtz eergy must be covex, we ca draw the commo taget dected by the red le betwee 1 B = 9:64fm3 ad 47:54fm 3 marked by the red dots. Its clato s the ressure P = :165MeV=fm 3 the hase trasto, whch s show by the red horzotal le Fg. 1. The ressure s also determed from a crossg ot o the black curve Fg. 3, whch shows the Gbbs eergy er baryo as a fucto of ressure. The Gbbs eergy o the crossg ot s ts equlbrated value gaseous ad lqud hases. To the cotrary, the chemcal otetals ad are ot equlbrated. As a matter of fact the black sold curves Fgs. 4 ad 5, whch show the chemcal otetals as the fuctos of ressure, have dscotutes deoted by the vertcal red dashed les. We have = 99:5MeV ad = 923:5MeV gaseous hase but = 898:7MeV ad = 929:3MeV lqud hase. The black dashed curves corresod to the cocave of the Helmholtz eergy Fg. 2 ad so are ot hyscally realzed. Cosequetly, the Gbbs codto o the hase equlbrum s ot sats ed. So as to costruct the hyscally reasoable lqud-gas mxed hase to be cosstet wth the Gbbs codto, we have to calculate the followg 11th-rak smultaeous olear equatos. The four equatos of them determe the e ectve masses ad the vector otetals of roto ad eutro gaseous (g) = (g) S + h g h g h + 3 m h 3 m (g) h! h! m (g) h 3 h 3 m (g) = ; V (g) = (g) B + h g h v (g) + h 3 h 3 v (g) h! h! v (g) h 3 h 3 v (g) = ; (g) = (g) S + h g h m (g) + h 3 h 3 m (g) h! h! m (g) h 3 h 3 m (g) = ; (17) 4

5 K. V (g) = (g) B + h g h v (g) + h 3 h 3 v (g) h! h! v (g) h 3 h 3 v (g) = ; (18) where the su x g dcates the gaseous hase. It s oted aga that the mea- elds are exressed by the e ectve masses ad the vector otetals. Smlarly, the equatos for lqud hase (l) = (l) S + h l h l + h 3 m h 3 m (l) h! h! m (l) h 3 h 3 m (l) = ; V (l) = (l) B + h h l v (l) + h 3 h 3 v (l) h! h! v (l) h 3 h 3 v (l) = ; (l) = (l) S + h l h m (l) + h 3 h 3 m (l) h! h! m (l) h 3 h 3 m (l) = ; V (l) = (l) B + h h l v (l) + h 3 h 3 v (l) h! h! v (l) h 3 h 3 v (l) = : (22) The other two equatos determe the mxture of gas ad lqud the hase trasto: B = 1 2 (1 a) B = f g (g) B + (1 f g) (l) B ; (23) B = 1 2 (1 + a) B = f g (g) B + (1 f g) (l) B ; (24) where f g ( f g 1) s the rato of gas. The baryo destes gaseous ad lqud hases are Z1 B = (g) d 3 k h (2) 3 k m (g) ; v (g) ; ; T k m (g) ; v (g) ; ; T ; (25) 5

6 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter Z1 B = (l) d 3 k h (2) 3 k m (l) ; v (l) ; ; T k m (l) ; v (l) ; ; T : (26) The last equato moses the equlbrum codto o ressures gaseous ad lqud hases: 1 X 3 2 Z 1 =; = X Z 1 =; h dk k4 Ek k dk k4 E k m (g) 1 2 m2 h 2 g h k 1 2 m2 h 2 l m (l) ; v (g) ; ; T + k m (g) ; v (g) ; ; T 1 2 m2 h 3 2 g m2! h! 2 g m2 h 3 2 g ; v (l) ; ; T + k m (l) ; v (l) ; ; T 1 2 m2 h 3 2 l m2! h! 2 l m2 h 3 2 l : (27) Solvg Eqs. (15)-(24) ad (27), we have the e ectve masses ad the vector otetals of roto ad eutro each of hases, the chemcal otetals of roto ad eutro, ad the rato of gas (or lqud) the hase trasto. It s oted aga that the caocal esemble the chemcal otetals ad are determed for the de te baryo destes (23) ad (24). For umercal calculatos we have used the globally coverget Newto algorsm Ref. [13]. The tral values are easly foud from the corresodg results the commo taget method. The umercal results for a = :3 ad T = 13MeV are show by the blue curves Fgs I Fg. 1 the ressure, whch s the value of the rght or left had sde of Eq. (27), creases slowly the hase trasto from gas to lqud agast the costat ressure from the commo taget. The rego of hase trasto betwee B = :17fm 3 ad :19fm 3 s wder tha the oe betwee B = :21fm 3 ad :14fm 3 from the commo taget. Ths s due to the fact that the Helmholtz eergy Fg. 2 s covex ad mmzed so as to be lower tha the commo taget. It s oted that the mmzato of the Helmholtz eergy leads to the equlbrum codto o ressure ad chemcal otetals the sothermal rocess through the fudametal thermodyamc equato df = SdT P dv + P dn. The Gbbs eergy Fg. 3 s also mmzed as =; a fucto of ressure because of G=A = F=A + P= B. The chemcal otetals Fgs. 4 ad 5 become the cotuous fuctos of ressure. Fgure 6 shows the rato of lqud f l = 1 f g, the asymmetry of gas a g = a l = (l) B (l) B (l) B +(l) B the hase trasto. (g) B (g) B (g) B +(g) B ad the asymmetry of lqud Here we have to dscuss the order of the hase trasto. Because the black curve ad the blue dashed curve Fg. 3 coect smoothly o the ots marked by the blue dots, 6

7 K. Myazak has o dscotuty. Referece [5] therefore asserts that the lqud-gas hase trasto asymmetrc uclear matter s the secod order. However, the argumet s ot cosstet wth the Gbbs codto, to whch each of the chemcal otetals rather tha the Gbbs eergy s relevat. We see readly Fgs. 4 ad 5 =@P =@P have dscotutes at the startg ad edg ots the hase trasto. It s therefore cocluded agast Ref. [5] that the hase trasto s the rst order. The same cocluso s also derved Ref. [1] from a d eret ot of vew. It s oted further that the exstece of the mxed hase s trsc to the rst-order hase trasto. Next, we vestgate the deedece of lqud-gas hase trasto o the asymmetry. Fgure 7 shows the ressure-desty sotherms for several asymmetres. The dashed curves are surous results from Eq. (14). The hyscally realzed ressure are the sold curves from Eq. (27). For a :5 although the dashed curves do ot show the ature of va der Waals equato-of-state, the mmzato of the Helmholtz eergy stll leads to the lqud-gas hase trasto. The rst order hase trasto from gas to lqud exsts utl a = :64. O the other hads, from a = :65 to a = :67 there are uer lmts o baryo desty, above whch we d o solutos of Eqs. (15)-(24) ad (27). The absece of hase equlbrum was foud rst Ref. [6] ad was recovered the recet work [9]. It was due to the desty-deedet [6] or the mometum-deedet [9] e ectve NN teractos. (Precsely seakg, the treatmet of the desty deedet -meso coulg Ref. [6] was ot hyscally accurate because the so-called rearragemet e ects [14] were ot take to accout.) I the reset work t s due to the eld-deedet coulg costats (1), whch are derved orgally from the mometum deedet coulg costats [11] ad lead to e ectve desty deedeces. Although there are foud Refs. [6] ad [9] the uer lmts o ressure, we have the uer lmts o baryo desty. The d erece s because Refs. [6] ad [9] vestgate the system that s ether caocal or gradcaocal whle we vestgate the caocal esemble. I ths resect t s also oted that the dscusso o the chemcal stablty Refs. [6] ad [9] s ot alcable to the caocal esemble the reset work. From a = :675 to a = :72 we see the retrograde codesato. For a examle Fg. 8 shows the rato of lqud as a fucto of ressure for the asymmetry a = :675 ad the temerature T = 13MeV. It decreases steely after the maxmum value s reached. Although the retrograde codesato s dscussed the other works [5,6,8,9], ts clear sgal, the varato of the rato of lqud, s reseted for the rst tme the reset aer. Above a = :72 we have o solutos of Eqs. (15)-(24) ad (27) at ay baryo desty ad so the uclear lqud does ot aear. The above results are summarzed Fg. 9, whch shows the secto of bodal surface. There s o crtcal ot, o whch the lqud brach coects wth the gas brach. The gas brach shows a backbed wth a bedg ot at a = :72, whch leads to the retrograde codesato. 7

8 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter We have develoed ew calculus of the lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter based o a ew RMF model Refs. [11] ad [12]. It s d eret from the geometrcal costructo used wdely the other works. The ressure ad the chemcal otetals are determed for de te values of the baryo destes ad the temerature. The ressure creases the hase trasto from gas to lqud agast the costat value from the Maxwell costructo of the hase equlbrum. The Helmholtz eergy s mmzed ad so s lower tha the commo taget. As a result the rego of the hase trasto s wder tha that from the Maxwell costructo. The chemcal otetals ad ad the dervatve of the Gbbs have o dscotutes. To the cotrary the dervatves of chemcal =@P =@P have dscotutes at the eds of the hase trasto. We ca therefore coclude agast the other works that the lqud-gas hase trasto asymmetrc uclear matter s the rst order. The lqud ad gas braches the secto of bodal surface are coected wth each other oly o the equal cocetrato a =, but there s o crtcal ot. Although the smlar result s foud usg the desty-deedet or the mometum-deedet NN teractos, we have co rmed that the absece of the crtcal ot s also reroduced usg the eld-deedet meso-ucleo coulg costats. Because the gas brach shows a backbed, the retrograde codesato of uclear lqud has bee roved. I a future work we wll aly the reset calculus to aother bary uclear system, that s, the strage baryo matter. 8

9 K. Myazak Refereces [1] J. Rchert ad P. Wager, Phys. Re. 35 (21) 1 [arxv:ucl-th/923]. [2] S.D. Guta, A.Z. Mekja ad M.B. Tsag, Advaces Nuclear Physcs, Vol. 26 (Kluwer Academc, 21) [arxv:ucl-th/933]. [3] V.E. Vola et al., Phys. Re. 434 (26) 1 [arxv:ucl-ex/6412]. [4] V.A. Karaukhov, Phys. Elem. Part. Atom. Nucl. 37 (26) 312 [htt://www1.jr.ru/pea/pea_dex.html]. [5] H. Müller ad B. D. Serot, Phys. Rev. C 52 (1995) 272 [arxv:ucl-th/95513]. [6] W.L. Qa, R-K. Su ad P. Wag, Phys. Lett. B 491 (2) 9 [arxv:uclth/857]. [7] P.K. Pada, G. Kle, D.P. Meezes ad C. Provdêca, Phys. Rev. C 68 (23) 1521 [arxv:ucl-th/3645]. [8] P. Wag, D.B. Leweber, A.W. Thomas ad A.G. Wllams, Nucl. Phys. A 748 (25) 226 [arxv:ucl-th/4757]. [9] J. Xu, L-W. Che, B-A. L ad H-R. Ma, arxv:ucl-th/7285. [1] C. Duco, Ph. Chomaz ad F. Gulmell, Nucl. Phys. A 771 (26) 68 [arxv:uclth/51229]. [11] K. Myazak, Mathematcal Physcs Prert Archve (m_arc) [12] K. Myazak, Mathematcal Physcs Prert Archve (m_arc) [13] W.H. Press, S.A. Teukolsky, W.T. Vetterlg ad B.P. Flaery, Numercal Reces C, 2d edto, 1992, Cambrdge Uversty Press [htt:// [14] F. Ho ma, C.M. Kel ad H. Leske, Phys. Rev. C 64 (21)

10 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter.3 Asymmetry =.3 T = 13 MeV Pressure (MeV/fm 3 ) Desty (fm 3 ) Fgure 1: The ressure-desty sotherm for the asymmetry a = :3 ad the temerature T = 13MeV. The black curve s the result from Eq. (14). The red horzotal le s the ressure the hase trasto derved from the commo taget rescrto. The blue curve s the ressure from Eq. (27). The vertcal red ad blue dashed les dcate the regos of the hase trasto. 1

11 K. Myazak 92 Asymmetry =.3 T = 13 MeV Helmholtz eergy (MeV) ρ 1 (fm 3 ) Fgure 2: The Helmholtz eergy er baryo as a fucto of volume for the asymmetry a = :3 ad the temerature T = 13MeV. The black curve s the result corresodg to the black curve Fg. 1. The red le s the commo taget that cotacts wth the black curve o the two red dots. The blue dashed curve betwee the two blue dots s the mmzed Helmholtz eergy satsfyg the Gbbs codto o the lqud-gas hase equlbrum. 11

12 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter 92 Asymmetry =.3 T = 13 MeV Gbbs eergy (MeV) Pressure (MeV/fm 3 ) Fgure 3: The Gbbs eergy er baryo as a fucto of ressure for the asymmetry a = :3 ad the temerature T = 13MeV. The black curve s the result corresodg to the black curve Fg. 1. It has a crossg ot deoted by the red le. The blue dashed curve betwee the two blue dots s the result satsfyg the Gbbs codto o the lqud-gas hase equlbrum. 12

13 K. Myazak.97 Asymmetry =.3 T = 13 MeV.965 µ / Pressure (MeV/fm 3 ) Fgure 4: The roto chemcal otetal as a fucto of ressure for the asymmetry a = :3 ad the temerature T = 13MeV. The black dashed curve corresods to the cocave of the Helmholtz eergy Fg. 2 ad so s ot realzed. Cosequetly, the black sold curve has a dscotuty deoted by the vertcal red dashed le. The blue curve s the result satsfyg the Gbbs codto o the lqud-gas hase equlbrum. 13

14 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter µ / Asymmetry =.3 T = 13 MeV Pressure (MeV/fm 3 ) Fgure 5: The same as Fg. 4 but for the eutro chemcal otetal. 14

15 K. Myazak 1.8 Rato of lqud Asymmetry of gas Asymmetry of lqud Desty (fm 3 ) Fgure 6: The black curve s the rato of lqud f l = 1 f g as a fucto of the baryo desty the hase trasto betwee B = :17fm 3 ad :19fm 3. The red ad blue curves are the asymmetres of gas ad lqud, a g = (g) B (g) B +(g) B (g) B ad a l = (l) B (l) B +(l) B (l) B. 15

16 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter.4.3 T = 13 MeV a=.6 a=.5 Pressure (MeV/fm 3 ).2.1 a=.1 a=.3 a=.4 a= Desty (fm 3 ) Fgure 7: The ressure-desty sotherms at T = 13MeV for the asymmetres from a = :1 to a = :6. The dashed curves from Eq. (14) are surous whle the sold curves from Eq. (27) are realzed. 16

17 K. Myazak.3.25 Asymmetry =.675 T = 13 MeV Rato of lqud Retrograde codesato Pressure (MeV/fm 3 ) Fgure 8: The rato of lqud as a fucto of ressure for the asymmetry a = :675 ad the temerature T = 13MeV. 17

18 Lqud-gas hase trasto the caocal esemble of asymmetrc uclear matter.4.35 T = 13 MeV Pressure (MeV/fm 3 ) lqud gas Asymmetry Fgure 9: The secto of bodal surface at the temerature T = 13MeV. 18